Symmetry Considerations Research Articles

Synthesis, Molecular and Supramolecular Structures of Symmetric Dinuclear Cd(II) Azido Complex with bis-Pyrazolyl s-Triazine Pincer Ligand

A new dinuclear Cd(II)-azido complex of 2,4-bis(3,5-dimethyl-1H-pyrazol-1-yl)-6-methoxy-1,3,5-triazine (PMT) pincer ligand is synthesized. Its single crystal X-ray structure reveals the dinuclear [Cd(PMT)(Cl)(N3)]2 formula. The triclinic crystal parameters are a = 9.323(4) Å, b = 10.936(5) Å, c = 11.312(6) Å, α = 112.637(10)°, β = 104.547(11)° and γ = 105.133(10)° while V = 944.1(8) Å3. Due to symmetry considerations, the asymmetric unit comprises a half [Cd(PMT)(Cl)(N3)]2 formula. The Cd(II) is hexa-coordinated with one tridentate PMT ligand in a pincer fashion mode in addition to one terminal chloride and two azide ions bridging the two Cd(II) centers in double μ(1,1) bridging mode. Unusually, the Cd-N(s-triazine) bond is not the shortest among the Cd-N interactions with the PMT pincer ligand. The supramolecular structure of the dinuclear [Cd(PMT)(Cl)(N3)]2 formula is controlled by a significant amount of Cl…H (16.4%), N…H (25.3%), H…C (9.8%) and H…H (37.2%) interactions based on Hirshfeld surface analysis. Careful inspection of the shape index map reveals the presence of some weak π-π stacking interactions between the s-triazine and pyrazolyl moieties. The percentage of C…C contacts is 1.9% where the C2…C8 (3.462 Å) is the shortest while the centroid–centroid distance is 3.686 Å. Natural charge analysis describes the charge transferences from the ligand groups to the Cd(II), while and atoms in molecules (AIM) give an indication on the properties of the Cd-N and Cd-Cl bonds.

Quantized Nonlinear Transport with Ultracold Atoms

In this letter, we propose how to measure the quantized nonlinear transport using two-dimensional ultracold atomic Fermi gases in a harmonic trap. This scheme requires successively applying two optical pulses in the left and lower half-planes and then measuring the number of extra atoms in the first quadrant. In ideal situations, this nonlinear density response to two successive pulses is quantized, and the quantization value probes the Euler characteristic of the local Fermi sea at the trap center. We investigate the practical effects in experiments, including finite pulse duration, finite edge width of pulses, and finite temperature, which can lead to deviation from quantization. We propose a method to reduce the deviation by averaging measurements performed at the first and third quadrants, inspired by symmetry considerations. With this method, the quantized nonlinear response can be observed reasonably well with experimental conditions readily achieved with ultracold atoms.

Rogue Waves Generator and Chaotic and Fractal Behavior of the Maccari System with a Resonant Parametric Forcing

Using the Asymptotic Perturbation (AP) method we can find approximate solutions for the Maccari equation with a parametric resonant forcing acting over the frequency of a generic mode. Taking into account its nonlocal behavior and applying symmetry considerations, a system with two coupled equations for the phase and amplitude modulation can be obtained. The system can be solved, and we demonstrate the existence of a big modulation in the wave amplitude, producing a rogue waves train and, in this case, these waves are not isolated. We then obtain a rogue waves generator, being able of producing and controlling the rogue waves’ amplitude. Another important finding is the existence of chaotic or fractal solutions, because of the presence of an arbitrary function in the solution.

Current-Induced Magnetic Skyrmions with Controllable Polarities in the Helical Phase

We report the current-induced creation of magnetic skyrmions in a chiral magnet FeGe nanostructure by using in situ Lorentz transmission electron microscopy. We show that magnetic skyrmions with controllable polarity can be transferred from the helical ground state simply by controlling the direction of the current flow at zero magnetic fields. The force analysis and symmetry consideration, backed up by micromagnetic simulations, well explain the experimental results, where magnetic skyrmions are created because of the edge instability of the helical state in the presence of spin-transfer torque. The on-demand generation of skyrmions and control of their polarity by electric current without the need for a magnetic field will enable novel purely electric-controlled skyrmion devices.

Piecewise-linear embeddings of decussate extended θ graphs and tetrahedra.

An nθ graph is an n-valent graph with two vertices. From symmetry considerations, it has vertex-edge transitivity 1 1. Here, they are considered extended with divalent vertices added to the edges to explore the simplest piecewise-linear tangled embeddings with straight, non-intersecting edges (sticks). The simplest tangles found are those with 3n sticks, transitivity 2 2, and with 2⌊(n - 1)/2⌋ ambient-anisotopic tangles. The simplest finite and 1-, 2- and 3-periodic decussate structures (links and tangles) are described. These include finite cubic and icosahedral and 1- and 3-periodic links, all with minimal transitivity. The paper also presents the simplest tangles of extended tetrahedra and their linkages to form periodic polycatenanes. A vertex- and edge-transitive embedding of a tangled srs net with tangled and polycatenated θ graphs and vertex-transitive tangled diamond (dia) nets are described.

Bridging atomistic spin dynamics methods and phenomenological models of single-pulse ultrafast switching in ferrimagnets

We bridge an essential knowledge gap on the understanding of all-optical ultrafast switching in ferrimagnets, namely, the connection between atomistic spin dynamics methods and macroscopic phenomenological models. All-optical switching of the magnetization occurs after the application of a single femtosecond laser pulse to specific ferrimagnetic compounds. This strong excitation puts the involved degrees of freedom, electrons, lattice, and spins out-of-equilibrium between each other. Atomistic spin models have quantitatively described all-optical switching in a wide range of experimental conditions, while having failed to provide a simple picture of the switching process. Phenomenological models are able to qualitatively describe the dynamics of the switching process. However, a unified theoretical framework is missing that describes the element-specific spin dynamics as atomistic spin models with the simplicity of phenomenology. Here, we bridge this gap and present an element-specific macrospin dynamical model which fully agrees with atomistic spin dynamics simulations and symmetry considerations of the phenomenological models.

A group theoretic approach to model comparison with simplicial representations

The complexity of biological systems, and the increasingly large amount of associated experimental data, necessitates that we develop mathematical models to further our understanding of these systems. Because biological systems are generally not well understood, most mathematical models of these systems are based on experimental data, resulting in a seemingly heterogeneous collection of models that ostensibly represent the same system. To understand the system we therefore need to understand how the different models are related to each other, with a view to obtaining a unified mathematical description. This goal is complicated by the fact that a number of distinct mathematical formalisms may be employed to represent the same system, making direct comparison of the models very difficult. A methodology for comparing mathematical models based on their underlying conceptual structure is therefore required. In previous work we developed an appropriate framework for model comparison where we represent models, specifically the conceptual structure of the models, as labelled simplicial complexes and compare them with the two general methodologies of comparison by distance and comparison by equivalence. In this article we continue the development of our model comparison methodology in two directions. First, we present a rigorous and automatable methodology for the core process of comparison by equivalence, namely determining the vertices in a simplicial representation, corresponding to model components, that are conceptually related and the identification of these vertices via simplicial operations. Our methodology is based on considerations of vertex symmetry in the simplicial representation, for which we develop the required mathematical theory of group actions on simplicial complexes. This methodology greatly simplifies and expedites the process of determining model equivalence. Second, we provide an alternative mathematical framework for our model-comparison methodology by representing models as groups, which allows for the direct application of group-theoretic techniques within our model-comparison methodology.

Direct statistical simulation of the Busse annulus

We consider direct statistical simulation (DSS) of a paradigm system of convection interacting with mean flows. In the Busse annulus model, zonal jets are generated through the interaction of convectively driven turbulence and rotation; non-trivial dynamics including the emergence of multiple jets and bursting ‘predator–prey’ type dynamics can be found. We formulate the DSS by expanding around the mean flow in terms of equal-time cumulants and arrive at a closed set of equations of motion for the cumulants. Here, we present results using an expansion terminated at the second cumulant (CE2); it is fundamentally a quasilinear theory. We focus on particular cases including bursting and bistable multiple jets and demonstrate that CE2 can reproduce the results of direct numerical simulation if particular attention is given to symmetry considerations.

The World in the Wave Function

<i>The World in the Wave Function</i>

1D diffeomorphism invariant model of a free scalar relativistic particle: Supervariable and BRST approaches

We apply the supervariable approach to derive the proper quantum Becchi–Rouet–Stora–Tyutin (BRST) and anti-BRST symmetries for the 1D diffeomorphism invariant model of a free scalar relativistic particle by exploiting the infinitesimal classical reparameterization (i.e. 1D diffeomorphism) symmetry of this theory. We derive the conserved and off-shell nilpotent (anti-)BRST charges and prove their absolute anticommutativity property by using the virtues of Curci–Ferrari (CF)-type restriction of our present theory. We establish the sanctity of the existence of CF-type restriction (i) by considering the (anti-)BRST symmetry transformations of the coupled (but equivalent) Lagrangians and (ii) by proving the symmetry invariance of the Lagrangians within the framework of supervariable approach. We capture the nilpotency and absolute anticommutativity of the conserved (anti-)BRST charges within the framework of (anti-)chiral supervariable approach (ACSA) to BRST formalism. One of the novel observations of our present endeavor is the derivation of CF-type restriction by using the modified Bonora–Tonin (BT) supervariable approach (while deriving the (anti-)BRST symmetries for the target space–time and/or momenta variables) and by symmetry considerations of the Lagrangians of the theory. The rest of the (anti-)BRST symmetries, for the other variables, are derived by using the newly proposed ACSA. We also demonstrate the existence of CF-type restriction in the proof of absolute anticommutativity of the (anti-)BRST charges.

High-order harmonic generations in tilted Weyl semimetals

We investigate high-order harmonic generations (HHGs) under comparison of Weyl cones in two types. Due to the hyperboloidal electron pocket structure, strong noncentrosymmetrical generations in high orders are observed around a single type-II Weyl point, especially at zero frequency. Such a remarkable DC signal is proved to have attributions from the intraband transition after spectral decomposition. Under weak pulse electric field, the linear optical response of a non-tilted Weyl cone is consistent with the Kubo theory. With extensive numerical simulations, we conclude that the non-zero chemical potential can enhance the even-order generations, from the slightly tilted system to the over-tilted systems. In consideration of dynamical symmetries, type-I and type-II Weyl cones also show different selective responses under the circularly polarized light. Finally, using a more realistic model containing two pairs of Weyl points, we demonstrate that paired Weyl points with opposite chirality can suppress the overall even-order generations.

Development of a new [formula omitted]–[formula omitted] angular correlation analysis method using a symmetric ring of clover detectors

A new method for γ–γ angular correlation analysis using a symmetric ring of HPGe clover detectors is presented. Pairwise combinations of individual crystals are grouped based on the geometric properties of the spectrometer, constrained by a single variable parameterization based on symmetry considerations. The corresponding effective interaction angles between crystal pairs, as well as the attenuation coefficients are extracted directly from the measured experimental data. Angular correlation coefficients, parameter uncertainties and parameter co-variances are derived using a Monte-Carlo approach, considering all sources of statistical uncertainty. The general applicability of this approach is demonstrated by reproducing known multipole mixing ratios in 177Hf, 152Gd and 116Sn, populated by either β-decay or (n, γ)-reactions, measured at the Institut Laue-Langevin, using the EXILL&FATIMA spectrometer and different configurations of the FIPPS instrument. The derived mixing ratios are in excellent agreement with adopted literature values with comparable or better precision.

Spin rectification effects in ferromagnetic metal microstrips induced by anisotropic magnetoresistance, planar Hall effect, and anomalous Hall effect

Spin rectification effect (SRE) is an important electrical detection tool employed to probe magnetic dynamics and spin-orbit torques, and it is also closely related to spin pumping measurements. The SRE has multiple physical origins, and the entanglement among them makes it difficult to derive quantitative information. Herein, we present a systematic study of the SRE in ${\mathrm{Co}}_{40}{\mathrm{Fe}}_{40}{\mathrm{B}}_{20}$ and Ni microstrips with out-of-plane-dominant microwave magnetic field excitation. The SREs from different physical mechanisms can be quantitatively decomposed via a symmetry consideration. The validity of the method is further supported by angular-dependent analyses. We observe both the anisotropic magnetoresistance induced SRE from the longitudinal induction current, and the planar Hall effect as well as the anomalous Hall effect induced SREs from the transverse induction current. Our work provides a comprehensive understanding of the SREs in ferromagnetic metal microstrips, and it is beneficial for quantitative analyses in microwave-related studies in spintronics.

Phase control of localization in the nonlinear two-mode system from harmonic mixing driving: Perturbative analysis and symmetry consideration

In this paper, we present a rigorous analysis of symmetry and underlying physics of the nonlinear two-mode system driven by a harmonic mixing field, by means of multiple scale asymptotic analysis method. The effective description in the framework of the second-order perturbative theory provides an accurate picture for understanding the Floquet eigenspectrum and dynamical features of the nonlinear two-mode system, showing full agreement with the prediction of symmetry considerations. We find that two types of symmetries play significant role in the dynamical features of this model, the mechanism behind which can be interpreted in terms of the effective description. The results are of relevance for the phase control of the atomic localization in Bose–Einstein condensates or switch of the optical signals in nonlinear mediums.

Gravitational wave modes in matter

A general linear gauge-invariant equation for dispersive gravitational waves (GWs) propagating in matter is derived. This equation describes, on the same footing, both the usual tensor modes and the gravitational modes strongly coupled with matter. It is shown that the effect of matter on the former is comparable to diffraction and therefore negligible within the geometrical-optics approximation. However, this approximation is applicable to modes strongly coupled with matter due to their large refractive index. GWs in ideal gas are studied using the kinetic average-Lagrangian approach and the gravitational polarizability of matter that we have introduced earlier. In particular, we show that this formulation subsumes the kinetic Jeans instability as a collective GW mode with a peculiar polarization, which is derived from the dispersion matrix rather than assumed a priori. This forms a foundation for systematically extending GW theory to GW interactions with plasmas, where symmetry considerations alone are insufficient to predict the wave polarization.

A Parametric Resonance for the Nonlocal Hirota–Maccari Equation

The nonlocal Hirota–Maccari equation is considered when a parametric excitation is acting over the frequency of a generic mode. Using the well-known asymptotic perturbation (AP) method, two coupled equations for the amplitude and phase can be obtained. We discovered the existence of an infinite-period bifurcation when the parametric force increases its value. Moreover, symmetry considerations suggest performing a global analysis of the two couples, in such a way that we find an energy-like function and corroborate and verify the existence of this infinite period bifurcation.

Thermal Hall responses in frustrated honeycomb spin systems

We study geometrical responses of magnons driven by a temperature gradient in frustrated spin systems. While Dzyaloshinskii-Moriya (DM) interactions are usually incorporated to obtain geometrically nontrivial magnon bands, here we investigate thermal Hall responses of magnons that do not rely on the DM interactions. Specifically, we focus on frustrated spin systems with sublattice degrees of freedom and show that a nonzero Berry curvature requires breaking of an effective $PT$ symmetry. According to this symmetry consideration, we study the ${J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{2}^{\ensuremath{'}}$ Heisenberg models on a honeycomb lattice as a representative example and demonstrate that magnons in the spiral phase support the thermal Hall effect once we introduce a magnetic field and asymmetry between the two sublattices. We also show that driving the magnons by a temperature gradient induces spin current generation (i.e., magnon spin Nernst effect) in the ${J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{2}^{\ensuremath{'}}$ Heisenberg models.

Near horizon aspects of acceleration radiation of an atom falling into a class of static spherically symmetric black hole geometries

The near horizon aspects (and beyond) of a black hole metric, which belongs to a large class of static spherically symmetric black holes, are considered here. It has been realized recently that an atom falling into a black hole leads to the generation of acceleration radiation through virtual transitions. In recent studies, it has been argued that this acceleration radiation can be understood from the near horizon physics of the black hole. The near horizon approximation leads to conformal symmetry in the problem. We go beyond the near horizon approximation in our analysis. This breaks the conformal symmetry associated with the near horizon physics of the black hole geometry. We observe that even without the consideration of the conformal symmetry, the modified equivalence relation holds. Further, our analysis reveals that the probability of virtual transition retains its Planck like form with the amplitude getting modified due to the beyond near horizon approximation. For the next part of our analysis, we have observed the horizon brightened acceleration radiation entropy (HBAR) for a Garfinkle-Horowitz-Strominger (GHS) black hole. We observe that the HBAR entropy misses out on quantum gravity like corrections while considering the conformal case. However, such corrections emerge when the conformal symmetry gets broken in the beyond near horizon analysis.

Moment constraints in beams with discrete and continuous rotational symmetry

This study presents a general analysis of how the transverse rotational symmetry of a beam imposes equality constraints among transverse beam moments. Efficient analytic methods are developed to derive symmetry-imposed constraints among $k$th order moments in beams with continuous or $n$-fold rotational symmetry for arbitrary $k$ and $n$. The formalism also enables one to construct symmetry arguments based on how discrete and continuous rotational symmetries manifest themselves differently in terms of beam moments. Three case studies on beams with continuous, twofold, and threefold rotational symmetries are conducted. We prove that, regardless of their triangulation, beams with threefold rotational symmetry (e.g., from electron cyclotron resonance ion source) always have the same rms properties as beams with cylindrical symmetry. These counterintuitive results derived purely from symmetry considerations have clarified beam dynamics at the Facility for Rare Isotope Beams.

On the generalizability of artificial neural networks in spin models

The applicability of artificial neural networks (ANNs) is typically limited to the models they are trained with and little is known about their generalizability, which is a pressing issue in the practical application of trained ANNs to unseen problems. Here, by using the task of identifying phase transitions in spin models, we establish a systematic generalizability such that simple ANNs trained with the two-dimensional ferromagnetic Ising model can be applied to the ferromagnetic qq-state Potts model in different dimensions for q \geq 2q≥2. The same scheme can be applied to the highly nontrivial antiferromagnetic qq-state Potts model. We demonstrate that similar results can be obtained by reducing the exponentially large state space spanned by the training data to one that comprises only three representative configurations artificially constructed through symmetry considerations. We expect our findings to simplify and accelerate the development of machine learning-assisted tasks in spin-model related disciplines in physics and materials science.